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vline

  • Let \varepsilon > 0.

    \varepsilon is fixed We have \begin{aligned} X_t = \sqrt{\alpha_t}X_{t-1} + \sqrt{1-\alpha_t} \varepsilon_{t}. \end{aligned} To have the concepts of independence and noise, we need to have probability measures. In the following text, we use lowercase q(x) to denote the density of the probability measure \mathbf{Q} corresponding to the random variable X. Others (e.g., q(x_t),p_{\theta}(x_t)) are the same. (p_{\theta} corresponds to \mathbf{P}_{\theta}). We also use q(x_{0:t}) to denote the density of (X_0,X_1,\cdots,X_t):=X_{0:t} for the probability measure \mathbf{Q}. Others are the same.

    Suppose q_{\text{want}}(x_0) is the density of X_0 we want to pursue. We do not know what q_{\text{want}}(x_0) is. We only have some eligible images (discrete data) with mass function {\color{blue}{q(x_0)}}. When this discrete data large, q(x_0)\approx q_{\text{want}}(x_0) in some sense of distribution. Our goal is to find a density p(x_0) of X_0 such that p(x_0)\approx q_{\text{want}}(x_0) in some sense of distribution.

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ASD

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File ~/Library/r-miniconda-arm64/lib/python3.10/site-packages/dash/dash.py:1295, in Dash.dispatch(self=<dash.dash.Dash object>)
   1294 try:
-> 1295     cb = self.callback_map[output]
        output = 'my-slider-1.value'
        self.callback_map = {'my-graph.figure': {'inputs': [{'id': 'my-slider', 'property': 'value'}], 'state': [], 'outputs_indices': 0, 'inputs_state_indices': 0, 'long': None, 'output': <Output `my-graph.figure`>, 'raw_inputs': [<Input `my-slider.value`>], 'manager': None, 'allow_dynamic_callbacks': None, 'callback': <function update_figure at 0x137c4fac0>}}
        self = <dash.dash.Dash object at 0x134cbbfd0>
   1296     _allow_dynamic = cb.get("allow_dynamic_callbacks", False)

KeyError: 'my-slider-1.value'

The above exception was the direct cause of the following exception:

KeyError                                  Traceback (most recent call last)
KeyError: "Callback function not found for output 'my-slider-1.value', perhaps you forgot to prepend the '@'?"

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File ~/Library/r-miniconda-arm64/lib/python3.10/site-packages/dash/dash.py:1295, in Dash.dispatch(self=<dash.dash.Dash object>)
   1294 try:
-> 1295     cb = self.callback_map[output]
        output = 'my-slider-1.value'
        self.callback_map = {'my-graph.figure': {'inputs': [{'id': 'my-slider', 'property': 'value'}], 'state': [], 'outputs_indices': 0, 'inputs_state_indices': 0, 'long': None, 'output': <Output `my-graph.figure`>, 'raw_inputs': [<Input `my-slider.value`>], 'manager': None, 'allow_dynamic_callbacks': None, 'callback': <function update_figure at 0x137c4fac0>}}
        self = <dash.dash.Dash object at 0x134cbbfd0>
   1296     _allow_dynamic = cb.get("allow_dynamic_callbacks", False)

KeyError: 'my-slider-1.value'

The above exception was the direct cause of the following exception:

KeyError                                  Traceback (most recent call last)
KeyError: "Callback function not found for output 'my-slider-1.value', perhaps you forgot to prepend the '@'?"

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File ~/Library/r-miniconda-arm64/lib/python3.10/site-packages/dash/dash.py:1295, in Dash.dispatch(self=<dash.dash.Dash object>)
   1294 try:
-> 1295     cb = self.callback_map[output]
        output = 'my-slider-1.value'
        self.callback_map = {'my-graph.figure': {'inputs': [{'id': 'my-slider', 'property': 'value'}], 'state': [], 'outputs_indices': 0, 'inputs_state_indices': 0, 'long': None, 'output': <Output `my-graph.figure`>, 'raw_inputs': [<Input `my-slider.value`>], 'manager': None, 'allow_dynamic_callbacks': None, 'callback': <function update_figure at 0x137c4fac0>}}
        self = <dash.dash.Dash object at 0x134cbbfd0>
   1296     _allow_dynamic = cb.get("allow_dynamic_callbacks", False)

KeyError: 'my-slider-1.value'

The above exception was the direct cause of the following exception:

KeyError                                  Traceback (most recent call last)
KeyError: "Callback function not found for output 'my-slider-1.value', perhaps you forgot to prepend the '@'?"

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File ~/Library/r-miniconda-arm64/lib/python3.10/site-packages/dash/dash.py:1295, in Dash.dispatch(self=<dash.dash.Dash object>)
   1294 try:
-> 1295     cb = self.callback_map[output]
        output = 'my-slider-1.value'
        self.callback_map = {'my-graph.figure': {'inputs': [{'id': 'my-slider', 'property': 'value'}], 'state': [], 'outputs_indices': 0, 'inputs_state_indices': 0, 'long': None, 'output': <Output `my-graph.figure`>, 'raw_inputs': [<Input `my-slider.value`>], 'manager': None, 'allow_dynamic_callbacks': None, 'callback': <function update_figure at 0x137c4fac0>}}
        self = <dash.dash.Dash object at 0x134cbbfd0>
   1296     _allow_dynamic = cb.get("allow_dynamic_callbacks", False)

KeyError: 'my-slider-1.value'

The above exception was the direct cause of the following exception:

KeyError                                  Traceback (most recent call last)
KeyError: "Callback function not found for output 'my-slider-1.value', perhaps you forgot to prepend the '@'?"

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File ~/Library/r-miniconda-arm64/lib/python3.10/site-packages/dash/dash.py:1295, in Dash.dispatch(self=<dash.dash.Dash object>)
   1294 try:
-> 1295     cb = self.callback_map[output]
        output = 'my-slider-1.value'
        self.callback_map = {'my-graph.figure': {'inputs': [{'id': 'my-slider', 'property': 'value'}], 'state': [], 'outputs_indices': 0, 'inputs_state_indices': 0, 'long': None, 'output': <Output `my-graph.figure`>, 'raw_inputs': [<Input `my-slider.value`>], 'manager': None, 'allow_dynamic_callbacks': None, 'callback': <function update_figure at 0x137c4fac0>}}
        self = <dash.dash.Dash object at 0x134cbbfd0>
   1296     _allow_dynamic = cb.get("allow_dynamic_callbacks", False)

KeyError: 'my-slider-1.value'

The above exception was the direct cause of the following exception:

KeyError                                  Traceback (most recent call last)
KeyError: "Callback function not found for output 'my-slider-1.value', perhaps you forgot to prepend the '@'?"

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File ~/Library/r-miniconda-arm64/lib/python3.10/site-packages/dash/dash.py:1295, in Dash.dispatch(self=<dash.dash.Dash object>)
   1294 try:
-> 1295     cb = self.callback_map[output]
        output = 'my-slider-1.value'
        self.callback_map = {'my-graph.figure': {'inputs': [{'id': 'my-slider', 'property': 'value'}], 'state': [], 'outputs_indices': 0, 'inputs_state_indices': 0, 'long': None, 'output': <Output `my-graph.figure`>, 'raw_inputs': [<Input `my-slider.value`>], 'manager': None, 'allow_dynamic_callbacks': None, 'callback': <function update_figure at 0x137c4fac0>}}
        self = <dash.dash.Dash object at 0x134cbbfd0>
   1296     _allow_dynamic = cb.get("allow_dynamic_callbacks", False)

KeyError: 'my-slider-1.value'

The above exception was the direct cause of the following exception:

KeyError                                  Traceback (most recent call last)
KeyError: "Callback function not found for output 'my-slider-1.value', perhaps you forgot to prepend the '@'?"

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File ~/Library/r-miniconda-arm64/lib/python3.10/site-packages/dash/dash.py:1295, in Dash.dispatch(self=<dash.dash.Dash object>)
   1294 try:
-> 1295     cb = self.callback_map[output]
        output = 'my-slider-1.value'
        self.callback_map = {'my-graph.figure': {'inputs': [{'id': 'my-slider', 'property': 'value'}], 'state': [], 'outputs_indices': 0, 'inputs_state_indices': 0, 'long': None, 'output': <Output `my-graph.figure`>, 'raw_inputs': [<Input `my-slider.value`>], 'manager': None, 'allow_dynamic_callbacks': None, 'callback': <function update_figure at 0x137c4fac0>}}
        self = <dash.dash.Dash object at 0x134cbbfd0>
   1296     _allow_dynamic = cb.get("allow_dynamic_callbacks", False)

KeyError: 'my-slider-1.value'

The above exception was the direct cause of the following exception:

KeyError                                  Traceback (most recent call last)
KeyError: "Callback function not found for output 'my-slider-1.value', perhaps you forgot to prepend the '@'?"

CCC

Code

  • ASd
  • ASDASDASD

III

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Table of Contents

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Getting up

  • Turn off alarm
  • Get out of bed

asdadas

Breakfast

  • Eat eggs
  • Drink coffee

Note

asdasdsd

222222

Table of Contents

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Dinner

  • Eat spaghetti
  • Drink wine

Going to sleep

  • Get in bed
  • Count sheep

3333333

Table of Contents

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  • 3333333

Going to sleep

  • Get in bed
  • Count sheep